Exercise 8, § 3

Let $T$ be a topological space and $m$ a bounded, countably additive (cf. Exer. 1d)) mapping of $\mathscr{B}(T)$ into $\mathbf{R}$. Denote by $\mathscr{T}$ the set of Borel subsets $A$ of $T$ having the following property: for every $\varepsilon > 0$, there exist a closed set $F$ and an open set $U$ in $T$ such that $F \subset A \subset U$ and $|m|(B) < \varepsilon$ for every Borel subset $B$ of $U - F$ (in other words, $|m|(U - F) < \varepsilon$). Show that $\mathscr{T}$ is a tribe of subsets of $T$ (first observe that $T \in \mathscr{T}$ and that $\mathscr{T}$ is a clan; it then suffices to prove that if $(A_n)_{n \in \mathbf{N}}$ is a sequence of pairwise disjoint elements of $\mathscr{T}$, the set $A = \bigcup_{n \in \mathbf{N}} A_n$ belongs to $\mathscr{T}$; to this end, choose closed sets $F_n$ and open sets $U_n$ such that $F_n \subset A_n \subset U_n$ and $|m|(U_n - F_n) < \varepsilon/2^n$, then set $F = F_0 \cup \cdots \cup F_p$ for $p$ sufficiently large and $U = \bigcup_{n \in \mathbf{N}} U_n$).